Precision Geolocation of Moving or Fixed Transmitters Using Multiple Observers

ABSTRACT

A method of geolocating a stationary transmitter observed by a fixed receiver device and at least two receiver devices, at least one of the receiver devices moving includes obtaining wavelength-scaled phase difference measurements between pairs of receiver devices, and 10 obtaining a result lattice of possible locations of the transmitter, one location more probable than the remainder. A method of geolocating a moving transmitter observed by a plurality of fixed or nearly fixed receiver devices, and a moving receiver device, includes obtaining wavelength-scaled phase difference measurements from the plurality of fixed or nearly fixed receiver devices to obtain a shape of the transmitter trajectory, measuring the phase difference 15 between the moving receiver device and at least one of the plurality of fixed or nearly fixed receiver devices to obtain a phase error residual, and moving an estimated starting point of the transmitter to obtain a best-fit residual.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a divisional of U.S. patent application Ser. No.12/542,541, filed Aug. 17, 2009, which is a divisional of U.S. patentapplication Ser. No. 11/464,762, filed Aug. 15, 2006, the disclosure ofwhich is hereby incorporated by reference as set forth in its entiretyherein.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention relates in general to geolocation technology usingradio waves, and, more particularly, to a system and method of precisegeolocation of moving or fixed transmitters using multiple observers.

2. Description of the Prior Art

A technology known as Kinematic Ambiguity Resolution (KAR) has beendeveloped for precision tracking using global positioning system (GPS)receivers that receive signals from many GPS satellites. The fundamentalapproach is to use carrier phase observations of many GPS satellites todetermine the location of a static or moving GPS receiver. Underfavorable circumstances, the location of a moving GPS antenna can befixed to centimeter accuracy within seconds.

At a fundamental level KAR involves measuring range to GPS satellitesusing carrier phase. Extremely precise measurements can be made oncarrier phase, usually better than 1/100 of a wavelength. Unfortunatelythese phase measurements have an ambiguity as to which cycle one ismeasuring. For example the GPS C/A code has a wavelength of 19 cm. Onemight be able to measure the carrier phase and determine the exactlocation of the wave to 1 mm, but with an ambiguity of 19 cm. If oneknew the transmission time of the measured cycle, the location of thetransmitter, and all propagation effects then the phase measurementdetermines the range to the satellite with an ambiguity of 19 cm. So theGPS receiver's antenna phase center is known to be on one of manyspherical shells centered on the satellite.

Typically the time at the GPS receiver is not precisely known, so inorder to fix a three-dimensional position in space, one must measure thesignal from at least four GPS satellites. This allows solution of threeposition coordinates and time of the GPS receiver. When observing thetime of arrival of the PN signal modulated onto the carrier, this issufficient to fix a unique location. However this location accuracy islimited by the precision with which the modulation time of arrival canbe measured. And this is, in turn, limited by the bandwidth of themodulation. GPS signals have modulation bandwidths of 1 and 10 MHz, soone can make relatively accurate measurements of the location based ontiming the modulation alone. This allows one to estimate the location ofthe GPS antenna to the level of meters.

Making measurements on the carrier phase instead of the modulation hasthe potential of improving the position fix accuracy from meters tomillimeters. Unfortunately, because of the ambiguity in carrier phasemeasurement, there are a multiplicity of locations where the antennamight be. These possible locations form a lattice pattern with spacingtypically on the order of a wavelength apart. The details of thislattice vary depending upon the geometry of the GPS satellites. If onecan resolve which of the possible locations is the true location, thendetermining location to millimeter accuracies becomes merely a matter ofeliminating all the biases may creep into the entire system. Examples ofbiases include errors in satellite position, changes in the phase centerof the GPS antenna with angle, local multipath reflections, troposphericpropagation effects, and ionospheric propagation effects. We willdiscuss elimination of biases at a later time. For now we will focus onthe difficult problem of resolving the ambiguities.

If the GPS receiver is fixed, a possible way to resolve the ambiguitiesis to wait while a GPS satellite moves to present a different geometry.One might start with a lattice of possible GPS locations and withsuccessive measurements (with different GPS geometries) compute newlattice. The lattice point corresponding to the true location willremain static while the other points move. With enough satellitegeometry change, the true location will become apparent. Anotherapproach is to choose an army of lattice points that best fit theobserved data. Then one calculates the residual rms phase error, the rmsof (measured phase-calculated phase) summed over all satellites.Eventually the true location's rms phase error will be much better thanthe rms phase error of the incorrect ambiguities.

Unfortunately GPS satellites move slowly. Fortunately there are a lot ofGPS satellites, and each transmits two different wavelengths. Observingwith two wavelengths simultaneously allows one to eliminate ambiguities.In the case of GPS, the ambiguities of both carrier wavelengths arespaced at a distance where the shorter wavelength has exactly one morecycle than the longer wavelength. This distance, known as the wide-laneambiguity, is about 0.9 meter. Observing more than four GPS satellitesserves to improve precision. A fifth satellite with a good geometry willadd phase measurement data that matches the true location, and not manyothers. Unfortunately, measurement errors create uncertainty as towhether a particular ambiguity matches the data or not. Usually theprocedure is to determine the lattice of possible locations, andcalculate the rms phase error associated with each one. If one locationproduces much better rms errors than all the others (e.g., by a factorof three or more), then this location is deemed to be correct.

In order to successfully perform KAR, one must have a data set withlittle measurement error and bias. An approach to obtaining this dataset has been to place a second GPS receiver near to the location whereKAR is being performed. This receiver is used as a Reference station. Itobserves each of the satellites being collected by the GPS receiver andsaves measurements on the code and especially the carrier phase. Eventhough the two receivers are separated, many of the errors due tosatellite position, atmospheric propagation, and ionospheric propagationappear in common at both receivers. So by subtracting the Reference'sdata from the Rover's data, a new data set with less error is generated.With GPS, KAR works best when the reference receiver is within 10 km ofthe rover's position. As separation is increased, separate propagationpaths through different parts of the atmosphere and ionosphere reducesthe amount of error that is correctable.

Geolocating a non-cooperative transmitter observed by multiplecollectors differs significantly from the GPS problem previouslydescribed. In the first instance, a complication is presented by thefact that transmitter waveforms are commonly not designed for the taskat hand. Instead of measuring the phase of transmitter carriers lockedto stable Rb or Cs oscillators, one is faced with a signal that may bedrifting in frequency and that may not even have a carrier. In addition,the signal may be narrow-band modulated so that instead ofTime-Difference-of-Arrival (TDOA) measurements on the signalconstraining the search region needing to resolve possible ambiguitiesto perhaps 10 meters, the search region may be 1 km or more.

The typical approach to geolocating such a signal is to cross-correlatethe signal seen at different collectors. The peak of thecross-correlation function in TDOA is used to define hyperboloids onwhich the transmitter must lie. The rate of change in phase of thecross-correlation function's peak is used to define theFrequency-Difference-of-Arrival (FDOA). FDOAs also define surfaces onwhich the transmitter must lie if the transmitter isn't moving. However,if the transmitter is moving, then FDOA measures a composite of thetransmitter and collector velocity. It is commonly thought that phasevs. time measurements cannot determine a transmitter's location if it ismoving.

In light of the foregoing, a need exists for a precise system and methodof geolocating a transmitter which is observed by collector deviceswhich are fixed or moving. In addition, a need exists for a geolocationsystem and method which serves to alleviate the problems of geolocatinga non-cooperative transmitter as described.

SUMMARY OF THE INVENTION

In one embodiment, the present invention is a method of geolocating amoving transmitter at position {right arrow over (X)}_(o)(t) observed bya plurality (N) of moving collector devices at positions {right arrowover (X)}_(i)(t), comprising obtaining (N−1) wavelength-scaled phasedifference measurements taken from independent pairwise combinations ofthe plurality of moving collector devices at a plurality of discretetimes, the first measurement taken at k=0, according toΔΦ(k)_(ij)=Φ_(j)(k)−Φ_(i)(k), and Φ_(i)(k)=|{right arrow over(X)}_(i)(k)−{right arrow over (X)}_(o)(k)|/λ+Θ_(i), where i and j areeach one of the plurality of collectors (i,j=1, 2, . . . , N), Θ_(i) andΘ_(j) are unknown constant phase offsets of the plurality of movingcollector devices, λ is the transmitter wavelength, and k is thediscrete time, and solving for an unknown three-dimensional position inspace, or if the moving transmitter is known to be on a surface of or ata known altitude above the earth, solving for a position in twodimensions.

In another embodiment, the present invention is a method of geolocatinga stationary transmitter observed by at least two receiver devices, atleast one of the receiver devices moving, comprising calibrating each ofthe receiver devices to reduce unknown phase bias and location bias,obtaining wavelength-scaled phase difference measurements between pairsof receiver devices, and obtaining a result lattice of possiblelocations of the transmitter, one location more probable than theremainder.

In another embodiment, the present invention is a method of geolocatinga moving transmitter observed by a plurality of fixed or nearly fixedreceiver devices, and a moving receiver device, comprising obtainingwavelength-scaled phase difference measurements from the plurality offixed or nearly fixed receiver devices to obtain a shape of thetransmitter trajectory, measuring the phase difference between themoving receiver device and at least one of the plurality of fixed ornearly fixed receiver devices to obtain a phase error residual, andmoving an estimated starting point of the transmitter to obtain abest-fit residual.

In still another embodiment, the present invention is a method ofimaging and geolocating a plurality of fixed transmitters and multipathcomponents observed by two receivers, at least one of the two receiversmoving, comprising, for each possible transmitter location, performingcomplex-conjugate product calculations between the two receivers,summing the product calculations with range compensation based on theshape of a path of the moving receiver to form an image of the pluralityof fixed transmitters and multipath components. GPS is used to determinethe receiver location, if ambiguity exists as to which of multiple phasedetermined points the receiver is located. The precision shape of thereceiver path is determined (with a possible constant position offset)by selecting a likely location and projecting that location in timeusing carrier phase measurements without cycle slips.

BRIEF DESCRIPTION OF THE DRAWINGS

In order that the advantages of the invention will be readilyunderstood, a more particular description of the invention brieflydescribed above will be rendered by reference to specific embodimentsthat are illustrated in the appended drawings. Understanding that thesedrawings depict only typical embodiments of the invention and are nottherefore to be considered to be limiting of its scope, the inventionwill be described and explained with additional specificity and detailthrough the use of the accompanying drawings, in which:

FIG. 1 illustrates an example geolocation system including a singletransmitter being observed by multiple receivers; and

FIG. 2 illustrates a geolocation technique with an aircraft carrying tworeceivers, in range of a transmitter.

DETAILED DESCRIPTION OF THE DRAWINGS

Reference throughout this specification to “one embodiment,” “anembodiment,” or similar language means that a particular feature,structure, or characteristic described in connection with the embodimentis included in at least one embodiment of the present invention. Thus,appearances of the phrases “in one embodiment,” “in an embodiment,” andsimilar language throughout this specification may, but do notnecessarily, all refer to the same embodiment.

Furthermore, the described features, structures, or characteristics ofthe invention may be combined in any suitable manner in one or moreembodiments. In the following description, numerous specific details areprovided. One skilled in the relevant art will recognize, however, thatthe invention may be practiced without one or more of the specificdetails, or with other methods, components, materials, and so forth. Inother instances, well-known structures, materials, or operations are notshown or described in detail to avoid obscuring aspects of theinvention.

Turning to FIG. 1, a geolocation system 10 including a singletransmitter being observed by multiple receivers is shown. System 10includes transmitter 12, as well as receivers 14, 16, 18 and 20, whichreceive radio signals along paths 22, 24, 26 and 28 respectively. Anadditional N number of receivers can be positioned to receive signalsfrom transmitter 12. The range from the transmitter 12 to the i^(th)receiver can be denoted as ρ_(i). The time that it takes a radio wave topropagate from transmitter 12 to receiver i is

ρ_(i)/c,  (1)

where c=speed of propagation. Often the time-of-transmission of awaveform is unknown, so it is impossible to measure the time ofpropagation directly. But if the signal received at multiple locationsis appropriately time-tagged, subsequent signal processing can measurethe time-difference-of-arrival (TDOA) of signals at the receivers 14,16, 18, 20, . . . , N. This, in turn allows one to calculate the rangedifference between the j^(th) and i^(th) receivers as

Δρ_(ij)=ρ_(j)−ρ_(i).  (2)

Typically there are two quantities that can be measured, modulationenvelope and carrier phase. Although the propagation velocities of thesequantities can be different in dispersive media (e.g. the ionosphere),variations in propagation velocity can be compensated for in manypractical scenarios. Throughout the instant specification, propagationin a vacuum where both group and phase velocity are equal to the speedof light, c, is assumed. Measurements of when a specific feature of theenvelope arrives allows one to measure a “group” TDOA. One approach tomaking this measurement is to calculate a cross-ambiguity functionbetween signals from two receivers. This ambiguity function has a peakat the TDOA. At other times, the difference in phase between tworeceivers can be measured. On a narrowband signal, the difference inphase can typically be measured much more accurately than themeasurement of the envelope time of arrival of the signal. However, onecannot usually resolve which carrier cycle is being observed. An n-cycleambiguity in the measurement and one actually observes is described as

ΔΦij=n+(ρ_(j)−ρ_(i))/λ+φ_(ij)err,  (3)

where: n is an unknown integer, λ is the radio's wavelength, andφ_(ij)err is the composite of all other errors.

The problem of resolving n and minimizing other errors is extremelydifficult. It is common to solve this by measuring the frequencydifference of arrival (FDOA). This can be done by taking the derivativeof the phase difference ΔΦij with respect to time. Often one canguarantee that a differential phase cycle is not slipped during themeasurement interval. In this case n is constant and does not appear inthe derivative. Likewise, φ_(ij)err can be held constant or made veryslowly varying. When FDOA is measured in this manner, one has a veryprecise measurement of the slant velocity difference of the transmitterobserved at the two receivers:

$\begin{matrix}{{\frac{{\Delta\Phi}_{ij}}{t}\lambda} = {\frac{\rho_{j}}{t} - {\frac{\rho_{i}}{t}.}}} & (4)\end{matrix}$

This measurement can in turn be useful in fixing position.

A measurement of Δρ_(ij) with the location of the two receivers knownfixes the location of the transmitter to be on a hyperboloid surfacewith foci at the location of the two receivers. Each time that anotherreceiver is added into the observing complex, another hyperboloid isdetermined. So with simultaneous measurements from 4 receivers, theintersection of 3 hyperboloids determines a location in 3-dimensionalspace. Note that with 4 receivers, one can compute (4−1)!=6 differences,but only 3 are mathematically independent.

The problem is more complicated if phase difference is being measured.Measurement using a pair of receivers with an unknown carrier phaseambiguity, n, results in a series of possible hyperboloids on which thetransmitter must lie. If the signal has a wavelength of a meter, thenthese hyperboloids are as close as one-half (½) meter along the baselinebetween the two receivers. Separation grows to be much larger if thetransmitter is located many baseline lengths from the receivers or if itis off the ends of the receiver baseline. Each time an additionalreceiver is added to the observing mix, another hyperboloid isdetermined. Each of these additional hyperboloids also have ambiguitiesdue to carrier phase ambiguities. The net result is a lattice ofpossible locations where these multiple hyperboloids intersect. Thelocations in this lattice can be separated on the order of a wavelengthnear the receiver baselines. Thus for a 300 MHz radio transmitter with 1meter wavelength, the lattice spacing can be less than a meter in aregion bounded by all 4 receivers. Of course, this spacing changes withgeometry and becomes large if the transmitter is distant from a clusterof receivers.

A frustrating circumstance results where (neglecting biases) one mightfix a transmitter's position to a small fraction of a wavelength if onecould determine which of many possible ambiguous locations thetransmitter lies on. Simply adding more observers does not directlysolve the problem since each added observer also has an unknown cycleambiguity. Consequently, the ratio of unknown to known quantities in theequation does not improve. However if multiple differential phasemeasurements are made while an observer moves, then the situationimproves. As long as one does not slip a cycle between measurements, theunknown integer, n, is constant and adding another phase differenceobservation does not increase the number of unknowns.

Consider the example of a fixed receiver 1 and a moving receiver j. Iftwo phase difference measurements are made at time t=1 and t=2, thefollowing measurement-related equations result after scaling themeasured phase difference in cycles by the wavelength, λ:

Δρ_(1j)(1)=ρ_(j)(1)−ρ₁(1)+n _(j)λ+φ_(1j)err(1)λ  (5)

Δρ_(1j)(2)=ρ_(j)(2)−ρ₁(2)+n _(j)λ+φ_(1j)err(2)λ  (6)

If we subtract these two measurements, if the location of thetransmitter and receiver 1 are constant, and if the phase error termsare constant, we obtain:

ΔΔρ_(1j)(2,1)=ρ_(j)(2)−ρ_(j)(1)  (7)

This result of differencing the wavelength-scaled phase difference is tofix a hyperboloid on which the transmitter must lie with one foci wherereceiver j was at time 1 and another foci where receiver j was at time2. If the measurements are very closely spaced in time, and the motionis small relative to the transmitter range, this reduces to the commonequation for FDOA where:

FDOA=ΔΔρ _(1j)(2,1)/(λ(t ₂ −t ₁))  (8)

and the hyperboloid can be approximated by a cone with its apex at theaverage receiver position. This cone is rotated about the velocityvector, {right arrow over (v)}, and has a half angle, Ψ. This cone isthe solution to the equation:

FDOA=(|{right arrow over (v)}|/λ)cos(Ψ).  (9)

If successive measurements are made an aperture can be synthesized alongthe flight path with wavelength-scaled phase measurements relative tothe starting point of ΔΔρ_(1j)(2,1), ΔΔρ_(1j)(3,1), ΔΔρ_(1j)(4,1),ΔΔρ_(1j)(5,1), ΔΔρ_(1j)(6,1), etc. Since none of these measurements addsan unknown, it might seem that only ΔΔρ_(1j)(2,1), ΔΔρ_(1j)(3,1), andΔΔρ_(1j)(4,1) (formed by 4 wavelength-scaled phase differencemeasurements) would be necessary to solve for the location of an unknowntransmitter. This might be true if the receiver were moved through alarge geometrical change in two or three dimensions. However if thereceiver motion is constrained to a line, measurements cannot solve fora rotational symmetry about the line. Furthermore, if the measurementsare made in rapid succession with small receiver motion compared to thedistance to the transmitter, the location of the transmitter along arange line projected from the moving receiver j will be determined muchpoorer than location in the cross-range direction. For these reasons, itis sometimes desirable to have more than one moving receiver in order toobtain a rapid and accurate fix on the transmitter's position.

The foregoing paragraphs described how to determine a location usingphase measurements. Determining the location was not predicated uponresolving the integer phase ambiguities between widely separatedreceivers. In fact, locations can be determined even if the receivershave unknown non-integer (but constant) biases in phase.

However, with a sufficient number of observers and proper geometrychange, the location accuracy derived from the techniques describedabove can be good enough to resolve integer ambiguities and to calibrateadditional unknown biases. Consider the simple planar example 30 in FIG.2. An aircraft carrying receivers 14,16 is moving along a straight line.The aircraft is in range of transmitter 12 as depicted. At time t=0 theaircraft deploys fixed receiver 14. Prior to deployment of receiver 14,the receivers 14, 16 are calibrated so that there is negligible unknownphase bias, φ₁₂err . At t=1 when physical separation is relatively smallthe first phase difference measurement is made. Because the initialseparation is small, the physical distance between integer phaseambiguities is very large. Therefore, relatively coarse positionmeasurements will resolve which integer ambiguity the transmitter 12 islocated on. As differential phase measurements continue while receiver16 flies away from receiver 14, two results occur. First, the accuracyof the hyperbolic surface's position increases as the baselineincreases. Second, the spacing between potential integer ambiguitiesbecomes smaller. As long as there is not a cycle slip, the decrease inambiguity spacing doesn't affect the ability to resolve the ambiguities.

With two receivers 14, 16 separated 0.1 kilometer and with a λ=1 metertransmitter 12, ambiguities perpendicular to the two-receiver baselineare spaced 0.01 radians. At a range of 10 kilometers, an unknown integercycle difference results in an ambiguity separated by 100 meters. Whenreceiver 16 has moved so that the baseline is 1 kilometer, the distantambiguities are now spaced 10 meters apart. Then if an additionalmeasurement is taken when the receivers are separated by 2 km, theambiguities are spaced 5 meters apart and intersect the original phaseline of position with an angle of roughly 0.1 radians or 6 degrees. AsFIG. 2 demonstrates, lines of constant phase move with receiver 16motion, except that the position of the correct ambiguity remains fixedwhile lines of constant phase pivot about it.

In the discussion thus far, is has been assumed that the transmitter 12is stationary. The concept of using a fixed receiver 14 with one (ormore) moving receivers 16, 18, etc., was described. It was shown thatdifferential phase measurements between a fixed and moving receivercould be processed so that integer ambiguities and phase biases betweenthe two receivers have no effect. It was also shown that, in terms ofresolution, the result is equivalent to a synthetic aperture lying alongthe trajectory of the moving receiver. There are multiple advantages toconceptualizing a synthetic aperture differential phase (DF). A primarymotivation is the ability to form an image to resolve more than onesimultaneous transmitter 12 or to resolve a direct path and multipathreflections. Resolving the phase ambiguity of widely spaced receiversmay lead to even more precise geolocation than possible using syntheticaperture techniques where phase ambiguities are not resolved.

For years FDOA techniques for geolocating transmitters have failedwhenever there is unknown transmitter motion. Depending on thevelocities and geometries of the receivers involved, even very smallmotion such as a ship steaming across the ocean or a person walkingthrough a park would introduce unacceptably large geolocation errors.Attempts to “fix” this usually involved modeling and making certainassumption about the transmitter motion. For example, one might model aship as having a constant course and speed and try to estimate theseparameters along with latitude and longitude. This approach is notsatisfactory in many situations, however.

The present invention teaches a different approach by adding enoughobservers so that the motion can be determined along with otherparameters. As noted earlier, an instantaneous measurement of TDOAdetermines range difference between two receivers 14, 16:Δρ_(ij)=ρ_(j)−ρ_(I) (refer to Equation 2). This, in turn, fixes ahyperboloid on which the transmitter 12 must lie with the foci being thelocations of the i^(th) and j^(th) receivers. Three independentdifference measurements can be made between pairs of four receivers, andthe intersection of these three hyperboloids can fix the transmitter'slocation in three dimensional space. Again, if the phase differencemeasurements have an ambiguity of one or more carrier cycles, a latticeof possible locations can be determined.

Consider the following example of a moving transmitter 12 which isobserved by five receivers (four fixed and one in motion). Assume that asequence of differential phase measurements are made by the fourreceivers without slipping any cycles. For each possible startinglocation, there is a unique differential phase bias constant consistingof an unknown integer ambiguity plus an unknown constant bias. For eachpossible starting location, the subsequent trajectory of the transmittercan be computed from the measured phase differences. If there are Npossible starting locations, there are N possible trajectories, one foreach assumed starting location. These starting locations might beambiguous positions on the lattice discussed above, or they might be alist of potential locations known from other outside information. Sincethe differential phase measurement is very precise (often a centimeteror better), the motion relative to each possible starting point may beextremely precisely known.

Suppose a fifth observer is added. It is mathematically possible thathaving a fifth observer will allow for a unique trajectory to bedefined. By way of example, consider a scenario where there are 4observers at great distance and a 5^(th) observer nearby. This might bethe situation when the transmitter 12 was known to be in a particularcity, and the 4 observers (e.g., receivers 14, 16, 18, 20) were ondistant satellites. If the satellites were sufficiently distant, rangechanges in the direction of a particular satellite would be essentiallyindependent of where in the city the transmitter 12 was. So the shape ofthe transmitter trajectory could be calculated based on observations ofthe 4 distant observers, and this trajectory shape would be virtuallyidentical regardless of where in the city the transmitter was.

A phase difference measurement could then be formed between the fifthreceiver and one of the original four. For each different location of atransmitter 12 starting point, the estimated trajectory shape is appliedto get an estimated trajectory. The range difference vs. time betweenthe fifth receiver and i^(th) receiver, ρ_(i5)(t), can then be computedbased on the transmitter trajectory. The residual difference between theobserved phase difference (in cycles) and the computed ρ_(i5)(t)/λ is anindication of the goodness of the estimate. This residual is minimizedwhen the best starting point is chosen. So by moving the hypotheticalstarting point in three dimensional space, the “best fit” residual canbe determined. The trajectory producing this “best fit” residual isassumed to be the best estimate of transmitter position and motion.

It should be pointed out that there are bad and good locations for thisfifth receiver. Positioning the fifth receiver at a large distance fromthe city would be bad since there would be only small differences in therange vs. time to it regardless of where in the city the transmitterstarting point was. On the other hand, a fixed receiver in the city nearthe trajectory of the transmitter would see a very large range/phasechange when the transmitter passed near to the receiver, and this wouldgreatly help constrain the transmitter's motion.

The key factor to accurate transmitter location with four receiversessentially fixed in the sky seems to be having large angular motionsbetween the transmitter and the remaining fifth receiver. Since onecannot rely on the transmitter always having large motion (and in factit might be stationary) it is normally best for receiver five to havesignificant motion.

If there is no transmitter motion, this degenerates to a situationidentical to the location results for the two-receiver syntheticaperture case with one fixed and one moving receiver. As previouslydiscussed in that case, the path covered by the moving receiver can beviewed as a synthetic aperture upon which the phase is known.

In the special case of four extremely distant receivers, one can apply atransform to make it identical to the case of a fixed transmitter andone moving receiver. This is done by first using the 4 distant receiversto calculate the transmitter's position vs. time:

{right arrow over (X)}(t)={right arrow over (X)} _(o) +{right arrow over(X)} _(R)(t)  (10)

where {right arrow over (X)}_(o)=Unknown initial transmitter position ina fixed coordinate system. {right arrow over (X)}_(R)(t)=Position vectorbetween initial position, {right arrow over (X)}_(o) and currentposition. This is the trajectory estimated from measurements on fourdistant receivers. Next, one can apply a coordinate transform to obtaina new coordinate system:

{right arrow over (y)}(t)={right arrow over (x)}(t)−{right arrow over(X)} _(R)(t)  (11)

In this coordinate system the transmitter is fixed at location {rightarrow over (Y)}_(o)={right arrow over (X)}_(o) that has yet to beestimated. The moving receiver five is at location {right arrow over(Y)}₅(t)={right arrow over (X)}₅(t)−{right arrow over (X)}_(r)(t), andthe formerly i^(th) fixed receiver is now at location {right arrow over(Y)}_(i)(t)={right arrow over (X)}_(i)−{right arrow over (X)}_(R)(t).

In order to make the analogy with one fixed and one moving receiver moreexact, one can now compute the motion of the i^(th) receiver imparted bymotion of the coordinate system origin, and remove it from any datameasurements made that involve the i^(th) receiver. Specifically, rangechange correction due to coordinate system motion is the dot product

{right arrow over (X)}_(R)(t)·{right arrow over (u)}_(i),  (12)

where {right arrow over (u)}_(i) is the unit vector in the direction ofthe i^(th) receiver.

In some situations, all of the receivers (e.g., 14, 16, 18, 20, 22) aremoving. An example would be when all receivers are on moving aircraft orsatellites. Consider a moving transmitter 12 at position {right arrowover (X)}_(o)(t) and moving receivers at positions {right arrow over(X)}_(i)(t) where i=1, 2, . . . , N The phase in wavelengths between thetransmitter at {right arrow over (X)}_(o)(t) and the output of thei^(th) receiver at {right arrow over (X)}_(i)(t) at time t=k is

Φ_(i)(k)=|{right arrow over (X)} _(i)(k)−{right arrow over(X)}_(o)(k)|/λ+Θ_(i),  (13)

where Θ_(i)=an unknown constant phase offset of the receiver and λ=thetransmitter wavelength.

Assuming that the transmitter 12 moves in three dimensions in a randomfashion, each time a measurement is made, there are three new unknownsrepresenting the change in transmitter location. Each receiver isassumed to have an unknown phase offset, so each receiver introduces aphase unknown the first time that it is used for a measurement. Howeversubsequent measurements with the same receiver do not introduce anyadditional unknowns as long as the phase offset is constant. It is wellknown that as long as equations are independent, it is possible to solvefor all the unknowns as long as the number of measurements are largerthan the number of unknowns.

In the case of an example geolocation technique having five receivers, afirst assumption can be that all receivers are phase locked together.One will only work with phase difference measurements between receivers.So with five receivers, one can make four independent phase differencemeasurements at each measurement time. It is assumed that the firstmeasurement is at t=0. At this time, one has three position unknowns andfour phase difference unknowns. The four phase difference unknowns arethe initial phase offsets between each of four receivers and anarbitrarily chosen fifth receiver. At t=1, four more measurements aretaken. There are three additional unknowns added, the position changefrom t=0 to t=1 in three dimensions. The phase difference betweenreceivers remains fixed, so no new unknowns are added by the receivers.

At each new measurement time, we add four measurements and threeposition-change unknowns. As a result, the number of measurements slowlyapproaches the number of unknowns. To illustrate, after the firstmeasurement at t=0 there are 3+4=7 unknowns and 4 measurements. Afterthe second measurement at t=1 there are (3+4)+3=10 unknowns and 4+4=8measurements. After the third measurement at t=2 there are (3+4)+3+3=13unknowns and 4+4+4=12 measurements. Finally, after the 4^(th)measurement there are (3+4)+3+3+3=16 unknowns and 4+4+4+4=16measurements. At this point if the measurements are sufficientlygeometrically diverse, it is possible to solve for the unknownthree-dimensional position after only four sets of measurements amongthe five receivers.

The same conclusion is reached when all five receivers are moving thatone reaches with four stationary receivers and one moving receiver.Again, in the former case, good geometrical situations one could solvefor a transmitter's location after motion in three directions. Startingwith a measurement at the start of each of the three motions and endingwith a measurement at the end of the last motion, a minimum of fourmeasurements are required. to define transmitter location at each of the4 measurement points.

In a similar manner, it is possible to make five independent phasedifference measurements between six receivers. After the firstmeasurement at t=0 there are 3 position unknowns and 5 phase differenceunknowns for a total of 8 unknowns. After the second measurement at t=1there are (3+5)+3=11 unknowns and 10 measurements. After the 3^(rd)measurement at t=2 there are (3+5)+3+3=14 unknowns and 15 measurements,and it is possible to solve for the unknowns with one extra degree offreedom (an overdetermined data set).

In a seven-receiver case we can make six independent pairs ofmeasurements. The first measurement at t=0 introduces (3+6)=9 unknownsand 6 measurements. The second measurement (3+6)+3+3=12 unknowns and 12measurements. So the location can be determined after measurements aremade at two times.

Two successive phase measurements may be regarded as a frequencymeasurement since frequency is rate of change of phase. As a result, theseven-receiver case of making two successive phase measurements betweensix independent pairs of receivers may be alternatively regarded asmaking six frequency difference measurements. Taking this view, sixindependent frequency difference measurements can be used to solve for 6unknowns. The unknowns are position (x,y,z) and velocity(v_(x),v_(y),v_(z)).

An important issue when using differential phase measurements istracking collector position extremely precisely. This issue especiallyarises when one of the collectors is an aircraft with a flight pathperturbed by turbulence. Consider the FIG. 2 example of an aircraftflying along a straight line. In practice the perturbed the flight pathis only approximately a straight line. If the true collector trajectorydeviations are precisely known, then one can straightforwardlycalculate, for a hypothetical transmitter position, how these trajectorydeviations affect the received signal's phase.

In order to calculate the transmitter's position based on differentialphase measurements, or equivalently image it's position, one must knowthe collector trajectory to a small fraction of the transmitter'swavelength. The same is true when estimating collector velocity in orderto geolocate using FDOA. It is important to note that for these purposesa constant bias error in collector position often has much less effectthan deviations in the collector path. The conclusion is that, forpurposes of FDOA, aperture synthesis, and related geolocationtechniques, GPS carrier phase measurements can make dramatic performanceimprovements without the need resolve all GPS carrier phase ambiguities.Consider the FIG. 2 example where phase change is observed while thecollection platform 12 moves over a baseline of 100 meters. If a 10 kmdistant 300 MHz transmitter signal's phase change is measured to 1/100of a cycle ( 1/100 of a meter with 1 meter wavelength), then thetransmitter location's cross-range error is 1/10000 of a radian or 1meter.

If the collector position undergoes an un-compensated 1 meter errortoward or away from the transmitter over these 100 meters, then theerror in transmitter position determination increases to 1/100 radiansor 100 meters in cross-range at a 10 km transmitter distance. If on theother hand the trajectory shape is known to 1/100 of a meter relativeaccuracy, then cross-range geolocation error is only the originalabsolute position error summed with another 1 meter cross-range error.Effects on measurement accuracy of transmitter range using phasecurvature across the synthesized aperture can be even more dramatic.

The conclusion is that in these situations the collector's trajectoryshape is more important than its absolute position. As a result it isoften not necessary to perform ambiguity resolution in GPS measurementsof the collector position to derive the benefits of GPS carrier phasemeasurements.

Another observation is that differential phase measurements on othernavigation signals such as Galileo or WAAS can also serve to makeprecise precision trajectory shape measurements. In fact, it is not evennecessary to use a navigation signal in order to improve the trajectoryshape as long as the transmitter location is known and the transmitteris simultaneously observed by both collectors.

A final important problem that has been solved is to calibrate drifts inthe reference oscillators at separated collectors. The usual approachwhen making precision measurements is to use Rb or even Cs referenceoscillators. But even these oscillators are not sufficiently accurateand stable for many aperture synthesis geolocation tasks. This isespecially true for aircraft oscillators that are subject toacceleration and temperature changes.

When observations of GPS carrier phase and/or other known-locationtransmitters form an overdetermined data set, the phase drift of thelocal oscillators can be measured. For example, consider the case ofwhere GPS carrier phase is used to track a collector's position changeover 10 minutes. If the collectors also have a view of WAAS, then thedrift in phase of the WAAS signal can be measured at each collector.When the calculated motion of the collector is removed from this dataset, the residual error is dominated by the oscillator phase drift andany errors in the WAAS signal. By this means the WAAS signal is used tocalibrate any oscillator phase drift. This means that we can also use aless expensive collector oscillator in many situations.

Finally, by differencing measurements on a WAAS signal in common viewfrom two collectors the WAAS signal's drift largely cancels leaving onewith even better differential phase measurements. This is of course amajor objective.

While one or more embodiments of the present invention have beenillustrated in detail, the skilled artisan will appreciate thatmodifications and adaptations to those embodiments may be made withoutdeparting from the scope of the present invention as set forth in thefollowing claims

1. A method of geolocating a stationary transmitter observed by a fixedreceiver device and at least two receiver devices, at least one of thereceiver devices moving, comprising: calibrating each of the receiverdevices to reduce unknown phase bias; obtaining wavelength-scaled phasedifference measurements between pairs of receiver devices; and obtaininga result lattice of possible locations of the transmitter, one locationmore probable than the remainder.
 2. The method of claim 1, wherein eachof the phase difference measurements between pairs is obtained withoutcarrier cycle slippage.
 3. The method of claim 1, wherein a motion pathof at least one moving receiver device is determined by measuring globalpositioning system (GPS) or global navigation satellite system (GNSS)carrier phase and solving using real-time kinematics (RTK) techniques.4. The method of claim 1, wherein a shape of a motion of at least onemoving receiver device is determined from carrier phase or differencedcarrier phase of global positioning system (GPS), global navigationsatellite system (GNSS), or known-location transmitters in common viewwithout resolving carrier phase ambiguity.
 5. The method of claim 1,wherein positions of multiple simultaneous energy sources are computedby summing range-corrected complex-conjugate multiplies of pairedsignals, the signals appropriately weighted by signal quality factors.6. The method of claim 5, further including forming images,multi-dimensional probability distributions, or other means to representthe positions of multiple simultaneous energy sources.